6z^2-5=17z

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Solution for 6z^2-5=17z equation:


Simplifying
6z2 + -5 = 17z

Reorder the terms:
-5 + 6z2 = 17z

Solving
-5 + 6z2 = 17z

Solving for variable 'z'.

Reorder the terms:
-5 + -17z + 6z2 = 17z + -17z

Combine like terms: 17z + -17z = 0
-5 + -17z + 6z2 = 0

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-0.8333333333 + -2.833333333z + z2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + -2.833333333z + 0.8333333333 + z2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + -2.833333333z + z2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + -2.833333333z + z2 = 0 + 0.8333333333
-2.833333333z + z2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
-2.833333333z + z2 = 0.8333333333

The z term is -2.833333333z.  Take half its coefficient (-1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
-2.833333333z + 2.006944445 + z2 = 0.8333333333 + 2.006944445

Reorder the terms:
2.006944445 + -2.833333333z + z2 = 0.8333333333 + 2.006944445

Combine like terms: 0.8333333333 + 2.006944445 = 2.8402777783
2.006944445 + -2.833333333z + z2 = 2.8402777783

Factor a perfect square on the left side:
(z + -1.416666667)(z + -1.416666667) = 2.8402777783

Calculate the square root of the right side: 1.685312368

Break this problem into two subproblems by setting 
(z + -1.416666667) equal to 1.685312368 and -1.685312368.

Subproblem 1

z + -1.416666667 = 1.685312368 Simplifying z + -1.416666667 = 1.685312368 Reorder the terms: -1.416666667 + z = 1.685312368 Solving -1.416666667 + z = 1.685312368 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + z = 1.685312368 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + z = 1.685312368 + 1.416666667 z = 1.685312368 + 1.416666667 Combine like terms: 1.685312368 + 1.416666667 = 3.101979035 z = 3.101979035 Simplifying z = 3.101979035

Subproblem 2

z + -1.416666667 = -1.685312368 Simplifying z + -1.416666667 = -1.685312368 Reorder the terms: -1.416666667 + z = -1.685312368 Solving -1.416666667 + z = -1.685312368 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.416666667' to each side of the equation. -1.416666667 + 1.416666667 + z = -1.685312368 + 1.416666667 Combine like terms: -1.416666667 + 1.416666667 = 0.000000000 0.000000000 + z = -1.685312368 + 1.416666667 z = -1.685312368 + 1.416666667 Combine like terms: -1.685312368 + 1.416666667 = -0.268645701 z = -0.268645701 Simplifying z = -0.268645701

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.101979035, -0.268645701}

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